Should I create the own faucet aerator

The screen is just one part of an aerator. There are a few different components to an aerator. It's best to get a new aerator instead of trying to track down all the individual pieces. It'll be worth the little bit of extra money. To make sure you get the right aerator the first time, I recommend that you bring the old one into the hardware store to match the threads.

That may be the rest of whatever, though I'm doubtful. But, the aerator will restrict flow, so run the faucet without the aerator at full force with the sink plugged for both hot & cold & see if anything noticeable flushes out. If it seems to clear out whatever then just put the aerator back on & see if it ever returns.

If nothing really produces from the unrestricted run, then it's probably a (or more) bad cone washer in your flexible supply lines under the sink. Those lines are only supposed to be very-very lightly wrenched on.

I suspect yours were way over tightened, which severely damaged the cone washers. They would be splayed out into the water stream & slightly eroded with every use of the faucet. If that's the case then it's nothing dangerous & just an avoidable nuisance.